Lemma REF  gives an upper bound of the Rademacher complexity based on the element-wise distance.
Notice that \(\left\Vert w^{(\infty )}\right\Vert  \le \left\Vert w^{(0)}\right\Vert  + \textsf {len} (w^{(0)}, \infty )\) .
To obtain a length-based generalization bound, we consider to use \(\mathcal {S}_{a, b}\)  to cover the length-constrained space \(\left\lbrace w : \left\Vert w\right\Vert  \le R\right\rbrace \) , and then taking a union bound.
For the \(\ell _2\)  ball covering number, we use the following result from [1]}.
